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International Contests
CentroAmerican
2010 CentroAmerican
2010 CentroAmerican
Part of
CentroAmerican
Subcontests
(6)
6
1
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Centroamerican Olympiad 2010, problem 6
Let
Γ
\Gamma
Γ
and
Γ
1
\Gamma_1
Γ
1
be two circles internally tangent at
A
A
A
, with centers
O
O
O
and
O
1
O_1
O
1
and radii
r
r
r
and
r
1
r_1
r
1
, respectively (
r
>
r
1
r>r_1
r
>
r
1
).
B
B
B
is a point diametrically opposed to
A
A
A
in
Γ
\Gamma
Γ
, and
C
C
C
is a point on
Γ
\Gamma
Γ
such that
B
C
BC
BC
is tangent to
Γ
1
\Gamma_1
Γ
1
at
P
P
P
. Let
A
′
A'
A
′
the midpoint of
B
C
BC
BC
. Given that
O
1
A
′
O_1A'
O
1
A
′
is parallel to
A
P
AP
A
P
, find the ratio
r
/
r
1
r/r_1
r
/
r
1
.
4
1
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Centroamerican Olympiad 2010, problem 4
Find all positive integers
N
N
N
such that an
N
×
N
N\times N
N
×
N
board can be tiled using tiles of size
5
×
5
5\times 5
5
×
5
or
1
×
3
1\times 3
1
×
3
. Note: The tiles must completely cover all the board, with no overlappings.
3
1
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Centroamerican Olympiad 2010, problem 3
A token is placed in one square of a
m
×
n
m\times n
m
×
n
board, and is moved according to the following rules: [*]In each turn, the token can be moved to a square sharing a side with the one currently occupied.[*]The token cannot be placed in a square that has already been occupied. [*]Any two consecutive moves cannot have the same direction.The game ends when the token cannot be moved. Determine the values of
m
m
m
and
n
n
n
for which, by placing the token in some square, all the squares of the board will have been occupied in the end of the game.
5
1
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Centroamerican Olympiad 2010, problem 5
If
p
p
p
,
q
q
q
and
r
r
r
are nonzero rational numbers such that
p
q
2
3
+
q
r
2
3
+
r
p
2
3
\sqrt[3]{pq^2}+\sqrt[3]{qr^2}+\sqrt[3]{rp^2}
3
p
q
2
+
3
q
r
2
+
3
r
p
2
is a nonzero rational number, prove that
1
p
q
2
3
+
1
q
r
2
3
+
1
r
p
2
3
\frac{1}{\sqrt[3]{pq^2}}+\frac{1}{\sqrt[3]{qr^2}}+\frac{1}{\sqrt[3]{rp^2}}
3
p
q
2
1
+
3
q
r
2
1
+
3
r
p
2
1
is also a rational number.
2
1
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Centroamerican Olympiad 2010, problem 2
Let
A
B
C
ABC
A
BC
be a triangle and
L
L
L
,
M
M
M
,
N
N
N
be the midpoints of
B
C
BC
BC
,
C
A
CA
C
A
and
A
B
AB
A
B
, respectively. The tangent to the circumcircle of
A
B
C
ABC
A
BC
at
A
A
A
intersects
L
M
LM
L
M
and
L
N
LN
L
N
at
P
P
P
and
Q
Q
Q
, respectively. Show that
C
P
CP
CP
is parallel to
B
Q
BQ
BQ
.
1
1
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Centroamerican Olympiad 2010, problem 1
Denote by
S
(
n
)
S(n)
S
(
n
)
the sum of the digits of the positive integer
n
n
n
. Find all the solutions of the equation
n
(
S
(
n
)
−
1
)
=
2010.
n(S(n)-1)=2010.
n
(
S
(
n
)
−
1
)
=
2010.